Boundary Conditions

4 In describing the boundary conditions (B.C.), we must note that:

1. Either we know the displacement but not the traction, or we know the traction and not the corresponding displacement. We can never know both a priori.

2. Not all boundary conditions specifications are acceptable. For example we can not apply tractions to the entire surface of the body. Unless those tractions are specially prescribed, they may not necessarily satisfy equilibrium.

5 Properly specified boundary conditions result in well-posed boundary value problems, while improperly specified boundary conditions will result in ill-posed boundary value problem. Only the former can be solved.

6 Thus we have two types of boundary conditions in terms of known quantitites, Fig. 9.1:

Figure 9.1: Boundary Conditions in Elasticity Problems

Displacement boundary conditions along T„ with the three components of ui prescribed on the boundary. The displacement is decomposed into its cartesian (or curvilinear) components, i.e.

uxi uy

Traction boundary conditions along r with the three traction components ti = njTij prescribed at a boundary where the unit normal is n. The traction is decomposed into its normal and shear(s) components, i.e tn,ts.

Mixed boundary conditions where displacement boundary conditions are prescribed on a part of the bounding surface, while traction boundary conditions are prescribed on the remainder.

We note that at some points, traction may be specified in one direction, and displacement at another. Displacement and tractions can never be specified at the same point in the same direction.

7 Various terms have been associated with those boundary conditions in the litterature, those are su-umarized in Table 9.1.

 u, r„ t, r Dirichlet Neuman Field Variable Derivative(s) of Field Variable Essential Non-essential Forced Natural Geometric Static

Table 9.1: Boundary Conditions in Elasticity

Table 9.1: Boundary Conditions in Elasticity

8 Often time we take advantage of symmetry not only to simplify the problem, but also to properly define the appropriate boundary conditions, Fig. 9.2.

 Gu Gt u x u y tn ts AB ? 0 ? 0 BC ? ? 0 0 CD ? ? a 0 DE 0 ? ? 0 EA ? ? 0 0

P y y 9 B Note: Unknown tractions=Reactions

P y y 9 B Note: Unknown tractions=Reactions y x

Figure 9.2: Boundary Conditions in Elasticity Problems

0 0